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Synchronization stability in simplicial complexes of near-identical systems

Fatemeh Parastesh1,2, Mahtab Mehrabbeik3, Karthikeyan Rajagopal1,4, Sajad Jafari3,5,6, Matjaž Perc7,8,9,10,11, Charo I. del Genio12,13,14,*, and Stefano Boccaletti12,13,15

  • *Contact author: charo.delgenio@trakia-uni.bg
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Phys. Rev. Research 7, 033039 – Published 8 July, 2025

DOI: https://doi.org/10.1103/ml7b-r35h

Abstract

Assessing the stability of synchronization is a fundamental task when studying networks of dynamical systems. However, this becomes challenging when the coupled systems are not exactly identical, as is always the case in practical settings. Here we introduce an extension of the Master Stability Function to determine near-synchronization stability within simplicial complexes of nearly identical systems coupled by synchronization-noninvasive functions. We validate our method on a simplicial complex of Lorenz oscillators, finding a good correspondence between the predicted regions of stability and those observed via direct simulation. This confirms the correctness of our approach, making it a valuable tool for the evaluation of real-world systems, in which differences between the constitutive elements are unavoidable.

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References (42)

  1. R. Albert and A.-L. Barabási, Statistical mechanics of complex networks, Rev. Mod. Phys. 74, 47 (2002).
  2. M. E. J. Newman, Structure and function of complex networks, SIAM Rev. 45, 167 (2003).
  3. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424, 175 (2006).
  4. S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, and M. Zanin, The structure and dynamics of multilayer networks, Phys. Rep. 544, 1 (2014).
  5. O. Artime, M. Grassia, M. De Domenico, J. P. Gleeson, H. A. Makse, G. Mangioni, M. Perc, and F. Radicchi, Robustness and resilience of complex networks, Nat. Rev. Phys. 6, 114 (2024).
  6. A. Pikovski, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
  7. S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, The synchronization of chaotic systems, Phys. Rep. 366, 1 (2002).
  8. S. Boccaletti, A. N. Pisarchik, C. I. del Genio, and A. Amann, Synchronization: From Coupled Systems to Complex Networks (Cambridge University Press, Cambridge, 2018).
  9. F. Varela, J.-P. Lachaux, E. Rodriguez, and J. Martinerie, The brainweb: Phase synchronization and large-scale integration, Nat. Rev. Neurosci. 2, 229 (2001).
  10. I. Franović and V. Miljković, Power law behavior related to mutual synchronization of chemically coupled map neurons, Eur. Phys. J. B 76, 613 (2010).
  11. T. Nishikawa and A. E. Motter, Comparative analysis of existing models for power-grid synchronization, New J. Phys. 17, 015012 (2015).
  12. H. Taher, S. Olmi, and E. Schöll, Enhancing power grid synchronization and stability through time-delayed feedback control, Phys. Rev. E 100, 062306 (2019).
  13. J. Boiko, I. Pyatin, O. Eromenko, and O. Barabash, Methodology for assessing synchronization conditions in telecommunication devices, Adv. Sci. Technol. Eng. Syst. J. 5, 320 (2020).
  14. D. N. Hajian, F. Parastesh, K. Rajagopal, S. Jafari, and M. Perc, When do chemical synapses modulate the formation of spiral waves? Nonlinear Dyn. 111, 22551 (2023).
  15. L. F. Lago-Fernández, R. Huerta, F. Corbacho, and J. A. Sigüenza, Fast response and temporal coherent oscillations in small-world networks, Phys. Rev. Lett. 84, 2758 (2000).
  16. I. V. Belykh, V. N. Belykh, and M. Hasler, Blinking model and synchronization in small-world networks with a time-varying coupling, Physica D 195, 188 (2004).
  17. A. E. Hramov and A. A. Koronovskii, Generalized synchronization: A modified system approach, Phys. Rev. E 71, 067201 (2005).
  18. J. Yao, H. O. Wang, Z.-H. Guan, and W. Xu, Passive stability and synchronization of complex spatio-temporal switching networks with time delays, Automatica 45, 1721 (2009).
  19. F. Li and X. Lu, Complete synchronization of temporal Boolean networks, Neural Netw. 44, 72 (2013).
  20. W. Zhang, C. Li, T. Huang, and J. Huang, Fixed-time synchronization of complex networks with nonidentical nodes and stochastic noise perturbations, Physica A 492, 1531 (2018).
  21. I. Franović, O. E. Omel'chenko, and M. Wolfrum, Bumps, chimera states, and Turing patterns in systems of coupled active rotators, Phys. Rev. E 104, L052201 (2021).
  22. F. Parastesh, S. Jafari, H. Azarnoush, Z. Shahriari, Z. Wang, S. Boccaletti, and M. Perc, Chimeras, Phys. Rep. 898, 1 (2021).
  23. I. Franović, S. Eydam, N. Semenova, and A. Zakharova, Unbalanced clustering and solitary states in coupled excitable systems, Chaos 32, 011104 (2022).
  24. A. Nazerian, S. Panahi, I. Leifer, D. Phillips, H. A. Makse, and F. Sorrentino, Matryoshka and disjoint cluster synchronization of networks, Chaos 32, 041101 (2022).
  25. A. Bayani, S. Jafari, H. Azarnoush, F. Nazarimehr, S. Boccaletti, and M. Perc, Explosive synchronization dependence on initial conditions: The minimal Kuramoto model, Chaos 169, 113243 (2023).
  26. L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett. 80, 2109 (1998).
  27. L. Huang, Q. Chen, Y.-C. Lai, and L. M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems, Phys. Rev. E 80, 036204 (2009).
  28. J. Sun, E. Bollt, and T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems, Europhys. Lett. 85, 60011 (2009).
  29. F. Sorrentino and M. Porfiri, Analysis of parameter mismatches in the master stability function for network synchronization, Europhys. Lett. 93, 50002 (2011).
  30. C. I. del Genio, M. Romance, R. Criado, and S. Boccaletti, Synchronization in dynamical networks with unconstrained structure switching, Phys. Rev. E 92, 062819 (2015).
  31. S. Coombes and R. Thul, Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function, Eur. J. Appl. Math. 27, 904 (2016).
  32. C. I. del Genio, J. Gómez-Gardeñes, I. Bonamassa, and S. Boccaletti, Synchronization in networks with multiple interaction layers, Sci. Adv. 2, e1601679 (2016).
  33. S. Panahi and F. Sorrentino, Group synchrony, parameter mismatches, and intragroup connections, Phys. Rev. E 104, 054314 (2021).
  34. C. I. del Genio, S. Faci-Lázaro, J. Gómez-Gardeñes, and S. Boccaletti, Mean-field nature of synchronization stability in networks with multiple interaction layers, Commun. Phys. 5, 121 (2022).
  35. F. Parastesh, K. Rajagopal, S. Jafari, M. Perc, and E. Schöll, Blinking coupling enhances network synchronization, Phys. Rev. E 105, 054304 (2022).
  36. S. Jafari, A. Bayani, F. Parastesh, K. Rajagopal, C. I. del Genio, L. Minati, and S. Boccaletti, Periodic systems have new classes of synchronization stability, Phys. Rev. Res. 6, 043105 (2024).
  37. F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874, 1 (2020).
  38. S. Boccaletti, P. De Lellis, C. I. del Genio, K. Alfaro-Bittner, R. Criado, S. Jalan, and M. Romance, The structure and dynamics of networks with higher order interactions, Phys. Rep. 1018, 1 (2023).
  39. L. V. Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Romance, R. Criado, M. Frasca, V. Latora, and S. Boccaletti, Stability of synchronization in simplicial complexes, Nat. Commun. 12, 1255 (2021).
  40. M. S. Anwar and D. Ghosh, Neuronal synchronization in time-varying higher-order networks, Chaos 33, 073111 (2023).
  41. See Supplemental Material at http://link.aps.org/supplemental/10.1103/ml7b-r35h for additional numerical validation using Lorenz oscillators and Rössler oscillators.
  42. https://codeberg.org/paraw/SimplicialMSF.

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Phys. Rev. Research 7, 033039– Published 8 July, 2025
  • Received 12 March 2025
  • Accepted 12 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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